Generalization of Markov Diophantine equation via generalized cluster algebra
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913425587699712 |
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| author | Gyoda, Yasuaki Matsushita, Kodai |
| author_facet | Gyoda, Yasuaki Matsushita, Kodai |
| contents | In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+ky^2z^2+2xz^2+2xy^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_10919 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generalization of Markov Diophantine equation via generalized cluster algebra Gyoda, Yasuaki Matsushita, Kodai Number Theory Combinatorics 11D25, 13F60 In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+ky^2z^2+2xz^2+2xy^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them. |
| title | Generalization of Markov Diophantine equation via generalized cluster algebra |
| topic | Number Theory Combinatorics 11D25, 13F60 |
| url | https://arxiv.org/abs/2201.10919 |